arXiv subjects
Publications and source records attributed to Emmanuele DiBenedetto.
Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.
The DeGiorgi classes $[DG]_p(E;γ)$, defined in (1.1)${}_{\pm}$ below encompass, solutions of quasilinear elliptic equations with measurable coefficients as well as minima and Q-minima of variational integrals. For these classes we present some new results (§2 and §3.1), and some known facts scattered in the literature (§3-§5), and formulate some open issues (§6).
For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$ (see (1.4) below). These minimizers arise also as {proper} solutions to the parabolic $1$-laplacian equation. Hence, the continuity condition continues to hold for such solutions (§ 3).
A Wiener-type condition for the continuity at the boundary points of Q-minima, is established, in terms of the divergence of a suitable Wiener integral.
For the logarithmically singular parabolic equation \[ u_t-Δ\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type estimate in the $L^1_{loc}$ topology, and we show that the solutions are locally analytic in the space variables and differentiable in time. The main assumption is that $\ln u$ possesses a sufficiently high degree of integrability, namely \begin{equation*} \ln u\in L^\infty_{loc}\big(0,T;L^p_{loc}(E)\big) \quad\text{for some} p\ge1. \end{equation*} These two properties are known for solutions of singular porous medium type equations ($0<m<1$), which formally approximate the logarithmically singular equation. However, the corresponding estimates deteriorate as $m\to0$. It is shown that these estimates become stable and carry to the limit as $m\to0$, provided the indicated sufficiently high order of integrability is in force. The latter then appears as the discriminating assumption between solutions of parabolic equations with power-like singularities and logarithmic singularities to insure such solutions to be regular.
Biophysical models are often populated by a large number of input parameters that are difficult to predict or measure experimentally. The validity and robustness of a given model can be evaluated by a sensitivity test to its input parameters. In this study, we performed local (based on a Taylor-like method) and global sensitivity (based on Monte Carlo filtering techniques) analyses of a previously derived PAR1-mediated activation model of endothelial cells. This activation model previously demonstrated that peptide-activated PAR1 has a different receptor/G-protein binding affinity that favors Galpha_q activation over Galpha_12/13 by approximately 800-fold. Interestingly, the present study shows that the parameter regulating the binding rate of activated PAR1 to Galpha_12/13 is indeed important to obtain the expected RhoGTP response. Moreover, we show that the parameters representing the rate of PAR1 deactivation and the rate of PAR1 binding to G_q, are the most important parameters in the system. Finally, we illustrate that the kinetic model considered in this study is robust and we provide complementary insights into the biological meaning and importance of its kinetic parameters.